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#72306752

order1024
nilpotency class6
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C26

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#512862
commutator, Frattini, cosocle#493C24
prime radical#61110C22
LCSG ▷ #493 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ #3151 ◁ G
LECSG ▷ #493 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1

1353 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 1353 conjugacy classes of subgroups — too many to draw. Only its 113 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degreeμ(G) = 64
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2,9,3)(4,15,22,10)(5,23,27,47,28,46,35,44)(6,17,32,12)(7,13,8,14)(11,40,42,31,16,26,24,57)(18,48,52,63,25,58,51,64)(19,37,30,43)(20,39,21,38)(29,34,45,33)(36,50,49,62,41,60,59,54)(53,55,61,56);(1,3,9,2)(4,15,22,10)(5,24)(6,29,32,45)(7,13,8,14)(11,27)(12,33,17,34)(16,35)(18,49)(19,53,30,61)(20,38,21,39)(23,57)(25,59)(26,44)(28,42)(31,46)(36,51)(37,55,43,56)(40,47)(41,52)(48,62)(50,64)(54,58)(60,63);(1,4)(2,10)(3,15)(5,25)(6,30)(7,21)(8,20)(9,22)(11,41)(12,43)(13,39)(14,38)(16,36)(17,37)(18,28)(19,32)(23,58)(24,49)(26,50)(27,51)(29,61)(31,54)(33,56)(34,55)(35,52)(40,60)(42,59)(44,63)(45,53)(46,48)(47,64)(57,62);(1,5)(2,11)(3,16)(4,18)(6,31)(7,27)(8,35)(9,28)(10,36)(12,44)(13,42)(14,24)(15,41)(17,47)(19,54)(20,51)(21,52)(22,25)(23,45)(26,33)(29,46)(30,62)(32,57)(34,40)(37,63)(38,59)(39,49)(43,64)(48,53)(50,55)(56,60)(58,61)

Polycyclic presentation:

2^10|a^2=ghi,b^2=i,e^2=h,g^2=i,h^2=j,c^a=ci,c^b=cij,d^a=de,d^b=df,d^c=dj,e^a=ej,e^b=eg,e^c=ej,e^d=ehj,f^a=fgh,f^b=fj,f^c=fj,f^e=fh,g^b=gi,g^d=gh,g^f=gi,h^b=hi,h^d=hj,h^f=hj,i^d=ij

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72306759, #72306836
maximal normal subgroup of#72306759, #72306836

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.7*F.8*F.9);;
gap> SetPower(coll, 2, F.9);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.9*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.8*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.7*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.8);;
gap> SetConjugate(coll, 7, 6, F.7*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 9, 4, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);